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Free time minimizers for the three-body problem

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Abstract

Free time minimizers of the action (called “semi-static” solutions by Mañe in International congress on dynamical systems in Montevideo (a tribute to Ricardo Mañé), vol 362, pp 120–131, 1996) play a central role in the theory of weak KAM solutions to the Hamilton–Jacobi equation (Fathi in Weak KAM Theorem in Lagrangian Dynamics Preliminary Version Number 10, 2017). We prove that any solution to Newton’s three-body problem which is asymptotic to Lagrange’s parabolic homothetic solution is eventually a free time minimizer. Conversely, we prove that every free time minimizer tends to Lagrange’s solution, provided the mass ratios lie in a certain large open set of mass ratios. We were inspired by the work of Da Luz and Maderna (Math Proc Camb Philos Soc 156:209–227, 1980) which showed that every free time minimizer for the N-body problem is parabolic and therefore must be asymptotic to the set of central configurations. We exclude being asymptotic to Euler’s central configurations by a second variation argument. Central configurations correspond to rest points for the McGehee blown-up dynamics. The large open set of mass ratios are those for which the linearized dynamics at each Euler rest point has a complex eigenvalue.

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Notes

  1. From Eq. (17) we get that \(D{\tilde{\nabla }} U\) equals the expression of Eq. (20) plus the term \(\nabla U (s) \otimes s^t M\) which we ignore since \(\langle s, \delta s \rangle _m =0\).

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Acknowledgements

Montgomery thankfully acknowledges support by NSF Grant DMS-20030177. Sanchez and Montgomery thankfully acknowledge support of UC-MEXUS Grant CN-16-78. Moeckel acknowledges NSF Grant DMS-1712656

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Richard Moeckel

  2. Department of Mathematics, University of California, Santa Cruz, CA, USA

    Richard Montgomery

  3. Instituto de Matemáticas, Universidad Nacional Autónoma de México, C. P. 04510, Ciudad de México, Mexico

    Héctor Sánchez Morgado

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  1. Richard Moeckel
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  2. Richard Montgomery
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  3. Héctor Sánchez Morgado
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Correspondence to Richard Montgomery.

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Moeckel, R., Montgomery, R. & Sánchez Morgado, H. Free time minimizers for the three-body problem. Celest Mech Dyn Astr 130, 28 (2018). https://doi.org/10.1007/s10569-018-9823-y

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